proof horizon

PROOF HORIZON // every note a theorem

euclid · E8 · shock · kerr

HORIZON · left/right on screen = left/right in your ears

E8 · 240 ROOTS · COXETER PLANE h=30

g(r) LEFT · ĝ(r) RIGHT · SHELLS AT r²=2n

EUCLID · 465 THEOREMS BY BOOK (ACROSS) AND STEPS FROM THE AXIOMS (DOWN)

SHOCK · WAVETABLE = LIVE PDE SOLUTION

Tap POWER. Each theorem Euclid proves becomes a chord that falls into the black hole.

PRESETS & MORPH

PRESET

MORPH

PROOF — Euclid's Elements, 475 nodes

ORDER Euclid's order, 1.1 to 13.18 layers out from the axioms proof cones, book by book

RATE

VOICES

LATTICE — x ∈ E8, one root per theorem

ROOT

VOICING shell radii √nharmonics nstacked fifths

STEP

RANGE

THETA BED

σ₃ CONTRAST

MAGIC — g left, ĝ right

SHELLS

TAIL

SIGN

SHOCK — ∂U/∂t + ∂F(U)/∂x = 0

SYSTEM linear advectioninviscid Burgersisothermal Euler Maxwell, perfectly hyperbolic (8 fields)relativistic Euler (Valencia) Bondi accretion (sonic horizon)radial wave in R⁸

FLUX Lax-FriedrichsRoe

LIMITER first orderminmodmonotonized centered superbeevan Leer

STRIKE SHAPE sinemagic function slicepulse / blastRiemann step E8 theta series E₄Michel inflow profile

STRIKE ON theorems citing axiomsevery theoremSTRIKE button only

CFL

CLEANING

EVOLVE

AMPLITUDE

SINE/PDE

HORIZON — Michel flow into Kerr

SPIN a/M

GAMMA Γ

TEMP c∞

START r₀

FALL

REDSHIFT

FRAME DRAG

SONIC CRACK

SPACE

WIDTH

ECHO TIME

ECHO FB

ECHO MIX

OUT

MODULATORS

LFO 1

  RATE

  DEPTH

LFO 2

  RATE

  DEPTH

MIDI — notes set ROOT

MOD WHEEL

PATCH

How it plays. The clock proves Euclid's theorems in turn. Each proof moves the 8-voice chord one root through E8 (theorems that cite axioms shift all eight voices a semitone; the rest move two voices a whole tone), re-strikes the PDE whose live solution is the waveform, and drops one note per reference into the black hole. Notes redshift with the lapse, swirl with frame dragging, crack where the inflow goes supersonic, and pass through the magic function's shells on the way in. Beyond the second shell g and ĝ are mirror images, so the two ears end in anti-phase.
Seven systems. Advection, Burgers and isothermal Euler, plus Maxwell's equations in perfectly hyperbolic form, where each of the eight voices plays its own field (E left, its partner right); relativistic Euler in Valencia form, sharing Γ with the black hole; Bondi accretion, where a disturbance struck inside the sonic point is carried in and cannot climb back out; and the radial wave equation in eight dimensions, released from the magic function itself with a wall at the last shell, so it rings at Bessel-zero ratios 1 : 1.53 : 2.04 : 2.54. The E8 theta series E₄ and the Michel inflow profile are strike shapes too.
Sources. Viazovska, arXiv:1603.04246 · Hariharan et al., arXiv:2604.23468 · Hariharan, MSci thesis (Imperial College) on the magic function · Gorard & Hakim, arXiv:2503.13877 · Gorard, arXiv:2402.02331 · Wolfram, arXiv:2107.07337. The Euclid graph matches Wolfram's published counts; apart from 1.5, 1.14 and 12.18 its edges are reconstructed, not Euclid's own.